Limit Calculator S Sub N

Limits of sequences are fundamental in calculus and analysis. This calculator helps you find the limit of a sequence sₙ as n approaches infinity. Learn how to determine whether a sequence converges or diverges.

What is a Limit of a Sequence?

The limit of a sequence sₙ as n approaches infinity (L = lim n→∞ sₙ) describes the value that the sequence approaches as it continues indefinitely. If the sequence gets arbitrarily close to L as n increases, we say the sequence converges to L.

Key concepts:

Convergent sequence: The sequence approaches a finite limit L.
Divergent sequence: The sequence does not approach any finite limit.
Bounded sequence: The terms of the sequence do not grow without limit.
Monotonic sequence: The sequence is either entirely non-increasing or non-decreasing.

Note: A sequence can only converge if it is bounded and monotonic.

How to Find the Limit of a Sequence

To find the limit of a sequence sₙ, follow these steps:

Check if the sequence is bounded (terms do not grow without limit).
Determine if the sequence is monotonic (always increasing or always decreasing).
If the sequence is both bounded and monotonic, it converges by the Monotone Convergence Theorem.
If the sequence converges, find the limit L by analyzing the behavior of sₙ as n increases.

If sₙ converges to L, then for any ε > 0, there exists an N such that for all n ≥ N, |sₙ - L| < ε.

Examples of Finding Limits

Example 1: Convergent Sequence

Consider the sequence sₙ = 1/n. Let's find lim n→∞ sₙ.

The sequence is bounded (0 < sₙ ≤ 1 for all n).
The sequence is monotonic (decreasing).
By the Monotone Convergence Theorem, the sequence converges.
As n increases, 1/n approaches 0.

Therefore, lim n→∞ 1/n = 0.

Example 2: Divergent Sequence

Consider the sequence sₙ = n. Let's find lim n→∞ sₙ.

The sequence is unbounded (grows without limit).
The sequence is monotonic (increasing).
Since the sequence is unbounded, it diverges.

Therefore, lim n→∞ n = ∞ (the sequence diverges to infinity).

FAQ

What is the difference between a limit of a sequence and a limit of a function?: The limit of a sequence sₙ as n approaches infinity describes the behavior of the sequence as it continues indefinitely. The limit of a function f(x) as x approaches a describes the behavior of the function near a point.
How do I know if a sequence converges?: A sequence converges if it is both bounded and monotonic. If the sequence is bounded and monotonic, it converges by the Monotone Convergence Theorem.
What does it mean for a sequence to diverge?: A sequence diverges if it does not approach any finite limit. This can happen if the sequence is unbounded or oscillates infinitely.
Can a sequence have more than one limit?: No, a sequence can only have one limit if it converges. If a sequence does not converge, it does not have a limit.